Presentation on theme: "Introduction to Macromolecular X-ray Crystallography"— Presentation transcript:
1 Introduction to Macromolecular X-ray Crystallography Biochem 300Borden LacyPrint and online resources:Introduction to Macromolecular X-ray Crystallography, by Alexander McPhersonCrystallography Made Crystal Clear, by Gale RhodesOnline tutorial with interactive applets and quizzes.Nice pictures demonstrating Fourier transformsCool movies demonstrating key points about diffraction, resolution, data quality, and refinement.Notes from a macromolecular crystallography course taught in Cambridge
2 Overview of X-ray Crystallography Crystal -> Diffraction pattern -> Electron density -> ModelResolution, Fourier transforms, the ‘phase problem’, B-factors,R-factors, R-free …
3 Diffraction: The interference caused by an object in the path of waves (sound, water, light, radio, electrons, neutron..)Observable when object size similar to wavelength.ObjectVisible light: nmX-rays: nm, 1-2 Å
5 Can we image a molecule with X-rays? Not currently.1) We do not have a lens to focus X-rays.Measure the direction and strength of the diffracted X-raysand calculate the image mathematically.2) The X-ray scattering from a single molecule is weak.Amplify the signal with a crystal - an array of orderedmolecules in identical orientations.
14 We are sampling the continuous transform at specific points determined by the periodic lattice. The lattice determines the spacing in the diffraction pattern.The intensity of the spots contains the information about the lattice content.Each individual spot on the diffraction pattern contains information about your entire molecule.
15 Diffraction pattern The intensity of each spot contains Practically:Assign a coordinate (h, k, l)and intensity (I) to every spot inthe diffraction pattern—Index and Integrate.Ihkl , shklThe intensity of each spot containsinformation about the entire molecule.The spacing of the spots is due tothe size and symmetry of your lattice.
17 Fourier transform:F(h)= ∫ f(x)e2πi(hx)dxwhere units of h are reciprocals of the units of xReversible!f(x)= ∫ F(h)e-2πi(hx)dh
18 Calculating an electron density function from the diffraction pattern F(h) = Fcos2π(uh+ a)F(h) = Fsin2π(uh + a)r(x) = ∫ F(h)e-2πi(hx)dhF = amplitudeu = frequencya = phaseExperimental measurements:Ihkl, shklFhkl ~ √Ihkl
19 Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect Methods
20 Heavy Atom Methods (Isomorphous Replacement) The unknown phase of a wave of measurable amplitude can bedetermined by ‘beating’ it against a reference wave of knownphase and amplitude.Combined WaveUnknownReference
21 Generation of a reference wave: Max Perutz showed ~1950 that a reference wave could be createdthrough the binding of heavy atoms.Heavy atoms are electron-rich. If you can specifically incorporate aheavy atom into your crystal without destroying it, you can use theresulting scatter as your reference wave.Crystals are ~50% solvent. Reactive heavy atom compounds can enterby diffusion.Derivatized crystals need to be isomorphous to the native.
23 The steps of the isomorphous replacement method
24 Heavy Atom Methods (Isomorphous Replacement) The unknown phase of a wave of measurable amplitude can bedetermined by ‘beating’ it against a reference wave of knownphase and amplitude.FPHFPFH and aHCan use the reference wave to infer aP. Will be either of two possibilities.To distinguish you need a second reference wave. Therefore, the techniqueis referred to as Multiple Isomorphous Replacement (MIR).
25 Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect Methods
26 Anomalous scatteringIncident X-rays can resonate with atomic electrons to result in absorptionand re-emission of X-rays.Results in measurable differences in amplitudeFhkl ≠ F-h-k-l
27 Advances for anomalous scattering methods Use of synchrotron radiation allows one to ‘tune’ the wavelength of theX-ray beam to the absorption edge of the heavy atom.Incorporation of seleno-methionine into protein crystals.
28 Anomalous scattering/dispersion in practice Anomalous differences can improve the phases in a MIR experiment (MIRAS)or resolve the phase ambiguity from a single derivative allowing for SIRAS.Measuring anomalous differences at 2 or more wavelengths around theabsorption edge: Multiple-wavelength anomalous dispersion (MAD).Advantage: All data can be collected from a single crystal.Single-wavelength anomalous dispersion (SAD) methods can work ifadditional phase information can be obtained from density modification.
29 Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect Methods
30 Molecular Replacement If a model of your molecule (or a structural homolog) exists, initialphases can be calculated by putting the known model into the unit cellof your new molecule.1- Compute the diffraction pattern for your model.2- Use Patterson methods to compare the calculated and measureddiffraction patterns.3- Use the rotational and translational relationships to orient the modelin your unit cell.4- Use the coordinates to calculate phases for the measured amplitudes.5- Cycles of model building and refinement to remove phase bias.
31 Direct MethodsAb initio methods for solving the phase problem either by findingmathematical relationships among certain phase combinations orby generating phases at random.Typically requires high resolution (~1 Å) and a small number of atoms.Can be helpful in locating large numbers of seleno-methionines for aMAD/SAD experiment.
32 Overcoming the Phase Problem Heavy Atom Methods (Isomorphous Replacement)Anomalous Scattering MethodsMolecular Replacement MethodsDirect MethodsF = amplitudeu = frequencya = phaseFTr(x,y,z)electron density
34 Are phases important?Duck intensitiesand cat phases
35 Does molecular replacement introduce model bias? Cat intensities withManx phases
36 An iterative cycle of phase improvement BuildingRefinementSolvent flatteningNCS averaging
37 Model buildingInteractive graphics programs allow for the creation of a ‘PDB’ file.Atom type, x, y, z, Occupancy, B-factor
38 The PDB File: ATOM 1 N GLU A 27 41.211 44.533 94.570 1.00 85.98 ATOM CA GLU AATOM C GLU AATOM O GLU AATOM CB GLU AATOM CG GLU AATOM CD GLU AATOM OE1 GLU AATOM OE2 GLU AATOM N ARG AATOM CA ARG AATOM C ARG AATOM O ARG AATOM CB ARG AATOM CG ARG AATOM CD ARG AATOM NE ARG AATOM CZ ARG AATOM NH1 ARG AATOM NH2 ARG A.
39 OccupancyWhat fraction of the molecules have an atom at this x,y,z position?B-factorHow much does the atom oscillate around the x,y,z position?Can refine for the whole molecule, individual sidechains, orindividual atoms. With sufficient data anisotropic B-factorscan be refined.
40 Least -squares refinement = S whkl (|Fo| - |Fc|)2hklApply constraints (ex. set occupancy = 1) and restraints (ex. specifya range of values for bond lengths and angles)Energetic refinements include restraints on conformational energies,H-bonds, etc.Refinement with molecular dynamicsAn energetic minimization in which the agreement between measuredand calculated data is included as an energy term.Simulated annealing often increases the radius of convergence.
41 R = Monitoring refinement S||Fobs| - |Fcalc|| S|Fobs| Rfree: an R-factor calculated from a test set that has not been usedin refinement.